Improvement in sinical quadrants



lation.

UNITED STATES PATENT Orricn.

ALEXANDER M. ci-nsnoL-n, or AN'r1eoN1sH',^NovA soo'riir,

IMPROVEMENT IN siNlcjAL QUADRANTS.

Specification forming part of Letters Patent No. 29,767, dated August'Et, 1860.

fo all whom, it may concern: s

Be it -known that I, ALEXANDER M. CHIS- HOLM, of'Antigonish, in the county of Sydney and Province of Nova Scot-ia, have invented a new and useful instrument by Whichmay.

be solved all useful problems in plane trigonometry and in navigatiouin all that relates to the keeping of the log, which linstrumentl denominate .Chisholms Mathematical Me- Chanical Scale`; and I do hereby declare that the following is a full. clear, and exact description of the same, reference being had to the accompanying drawing, which represents the instrument and forms pari of this specification.

The principal object of my invention is to obviate the necessity of calculation for keeping a-correct reckoning by the log, the coasterV or seaman having no knowledge whatever of sines and tangents being enabled by its aid to preserve as accurate a knowledge of his relative position as can be obtained by a-log. It will give a correct reckoning at sea, either by middle latitude or Mercator sailing; give the dierence of latitude and departure for every'point and quarter-point of the quadrant,

and for every degree turn departure into 1onigitude, either by middle latitude or Mercator potenuse is equal to that of both the base and the perpendicular.

The scale is a plane square of ten inches or other extent divided and subdivided into less squares of tenths each way, answering thepurpose of enumeration, numbered at the angle .of each division upon the sides, respectively,

` marked A B CD. There is a quadrant in one of the four' angles marked E, the radius of which extends to sixty (GO) of the subdivisions, or six-tenths of the length of the-sides, and this quadrant is graduated with th'c de grees, points, and quarter-points. In the vcrtexot' the angle agreeing with the quadrant is pivotcd by alpi-votv 1v-theL attached-index F. This index F consists of a stripe-r 'lata-marked to agree with the divisions onthe'isc'alel The graduated edge of this index` is used-'asia' hy! potenuse to any triangle that may-be for1ned-,. and its sixtieth division forms a tangent to the circle of which the quadrant'formsa parton Whatever part of the quadrant the index" is placed. It will also showthe secant to any degree, and thus themeridional parts answering to every degree of lthe quadrantlmaybe taken separately.lv The scale'andattachedfiirdex F serve to perform `arithmeltical-:compu tations andtosolve 'allprobl-'emsin 'right-ane gled trigonometry, and in planegparalleL-and Mercator sailing; but the assistance of the' detached index H is necessary torsolve problems in oblique-angled trigonometry.

The use ot the scale and attached index will be described before fur-therreference is' made to the assisting.detacl'rendindex` The use ofthe'iscale 'andi-index forl-aritiili inetical computations is astoll'owsz` twelfth smaller graduation on the 'index to the perpendicular 0l 10, or thefirst larger graduation'on the side A of the'scale. Supposing the numbers 10,20, 30, dac., shown to represent units, the perpendicular otv any mnltiplicand Will then intersect a graduation? of the index denoting the sum. y

Example: 1O 19=l20,or 5 12=`=6(); &c':, and the same for any other multiplier, always bringing the multiplier on the index-to the perpendicular of the-number l0 orl uniton? the sideA of the scale.

RULE Fon DivIsIoNf Suppose l2 to be the divisor. Adjust the index the same as before, and in contact With Athe dividend on thc index will "be found the' perpendicular of the quotient on thefside A.

Example: tlO-llz'i, or 72-1-12=6,-&c., the perpendicular in contact with any "divi-- dend on the index thus showing'the' quotient; on the side A ot the scale.

Foa THE Renn ory PRoPoRTIoN.

Example: As 50 is to 60, so is` 40 to a' de; sired tourtlrterni. Set 60 on the index to the perpendicular of'O on the side A of the scale,

and then' 40" on the index will give 48 on the scale. is 48.

In right-angled trigonometry all problems flfhetourth term or answer therefore i are solved bythe triangle'formed byv takingv one leg on the side A, and from the lineof.

the given quantity trace the other leg down- .lward to the line of its given quantity on B, f to which set the index, which will showr thequantity of the hypotenuse or longest side. This is applicable to all cases.

' CASE 1. I The side 'A equals eighty,"the side B equals sixty. Required; thehypotenuse On the side A.' (counting vfrom the pivot) take 80.

Trace thatgraduationdownward till you com e f.

to the grada ation marked 60 on the side B. 'lo that intersection set the index, an d the number' thereon will show the Ahypothen use to b e v100.

' v CASE -2.

CASE 3.

The hypotenuse equals one hundred.v The side A equals sixty. Required: the side B? Take 60 on the side A, bring 100 en index to meet that graduation continuedl downward and it will give the required side 8O on B, or takeI the given side on B and 100 brought to it Willugive the required side 80 on A.

AT. Rv-If the angle at a or the pivot were required in Cases 1 and 2, it would appear on the quadrant at the edge of the index to be thirty-seven degrees, but in Case 3 it would befifty-seven degrees.

In plane sailing the top or side A of the scale seems the most convenient to be used as a meridian, and will be so considered in the following operations, the difference of latitude either north or south to be count-ed from the pivot u, the distance being always taken or found onthe index.

CASE 1.

Course and distance given, to find the difference ot' latitude and departure.

`Example: A'ship sails south twenty-five degrees, easterly ninety-'six miles. Required: her difference of latitude and departure? -Ifule'v Set the index to the course twentylive degrees on the quadrant. From the distance', 96, trace the graduation nearest to that number o-n the index to the side A, and thence" to the pivot.` You'will then have 87 for the dierence of latitude;` alsoo'n the .samebetween the side A and the' edge of the index you will have l0-. l for theA departure.

` CASE 2. i

Course and diiference of latitude given, to find the distance and departure. l

Eample: A ship sails southwest by south till her dicrence of latitude is titty-eight miles. Required: her departure and dist ance? Rule:4 Set the index to the course three points from the meridian, or A. Take the difference of latitude 58. Trace the perpendicular downward to the edge ot' the index, which will be 70 for the distance. The graduation 58 from the index to the meridian will then quote thirty-eight miles and tenths for thev departure.

Another exmnple: A ship sails north thirtyeight degrees, westerly till she dit'rer her lati.- tude sixty miles. Required: her distanee'and departure? v Rule: Set the index to the course thirtyei-ght degrees. Take 60v on the, side A. 'Trace its perpendicularv down to the edge of the in-` dex and it will be the departure, 47. It will also meet the distance on the index-viz., 76.

CAsE 3.

Course and departure given, tofind distance and 'difference of latitude.

Example: A ship sails south five points,

' east till her departure is fifty-four miles. Re-

quired: the tude? Rule: Set the index to five points from the meridian, or A. On the side B take 5.4, thedeparture, and trace it toward the left till it meets the edge yof the index, where you will find 65 for the distance. Trace the division distance and dierenceoflatior igierpendicularv from that to the meridian and it will give difference of latitude 36;

CASE l.

there will show the course on the quadra-nt 57 o easterly.

CASE 5. l

Diderence of latitude and departure given, to find course and distance. .Erdmple A` ship sails between north and west till her dierence of latitude is forty miles and her departure fifty-five miles. Required: her course and distance? Rule: From u on A take the difference of latitude, forty miles; trace its perpendicular down to ftyQi-ive miles, the departure; set the index to the angle of meeting thereon; you will then'liave sixtyeight for the distance;

A will then be the departure, 4G miles, and the index kept 'the quadrant tor'thecours'e'. .1

- thirty-limites.Required Seti-it iithedeparture; 3(2,tna`cedfromthe side q showffth difference .of latitudeon the meridian-.or sideAf'to' be 45,.nearl.y;"the-i`ndex so :.:Rttlerlsetgthe index "tof-thehgiven latitude,

. ion enfuit-inherent by itwillfbe the difference itwi'llalse ...toten the. 'fifty-fuient .degree yoir' Distance.and-departure ,giveml to tind'the vcourse `and difference oflatitudei v ,.:Eamplef' AShiptsails.` tween the no1-th' and-.feastziftyffonr 'Iirile er departure" is.

: hel ourseand di-ter.-

enfcenf.latitude.tA

'.Takefthle'distanceff. on the index;

Beto .the edgejof ftheindexgi the perpendicu` lar,l tracedy 4from .theangle' .of. meeting, .will

set will' SliOiVjhe course on .ithe'quadrant toV be three points. f 1 Y,

Thus I havegoneove'r all the variety con- .CASE 1.V

Y Two places in` one parallel; their latitude and distance given, totind theirdifterenee of follows:

longitude.

y Eample: Two places lin latitu'de`50; distantseventy-six iniles. Required: their differenee:of-longiltude? onfth'e qnad'iant;'take tliejeasting or West ing, ',Ziglon thefs'ide A; trace its perpendicular dolnetofthe edgeof the index, and the divis- Two places in one parallel; their latitude and diterence of longitude given, to find their distance.

.Ewa-nipte: '.Two `places, in latitude 50; their diierence" of longitude 1181:Requi1edz their distanee.?l..` l

Ruta-:Setltlxe index tothe given latitude, 50, on the quadrant, and from 118 on it trace the perpendicular to the side A, Where you will tind 76 for the distance required. c

CASE

Two places in one latitude; their distance and diierence of longitude given, to find the. latitude they are in.

"Edmple: A ship sails due West two thou's` andseven hundred miles, and then nds her difference of longitude to'be four thousand two hundred minutes. Required: the latitude the ship sails in? In the above example the given numbers are too large for the limits of the scale; but there being an equal number of oiphers in both I take 42 andfan'd work with them,".

Rule; Take 2. on the indexand bringit to the twenty-seventh. perpendicular-on the The index will then fall. on the fiftietliv the course, dep

deg. ,ree on the Aquadrant,which is .the-latitude required.-`

OASE 4'.

south;their` 'distance in one parallel given, to tind their distance in another parallel.

- ExampZe:fSu'ppose two ships' in latitude 50 vnorth',distai`1t.tvvo 'liund're'd miles, sail 'both directlyl north intoflatitu'de 73. Required: their distance'in that parallel? Rule: Set the indexto Athe latitude first given, 50; for the distance, 200, take '20 on side of A, (assuming 10 to 1;)l observe what 'number on the index will come in Contact with the perpendicular of .20; here it will be, 3l; move the index up'to latitude 73;.When there, observe what perpendicular'will be again incontactwith 31011 the index; here "it Will be the ninth, (count-ing ten to 4one again you then have 90) or' 91: the' distance of they two ships in that parallel.

l OASE 5.

.Two ships in one parallel with their dis- Two ships in'vlatitude dis-1 -tant two hundred miles; sail both directly "l north tilltlieir. distance -is .-but ninety-one miles. Required: the latitude arrived a t.? 'Ruler Set the index as before, .and move the number 31011 the index to the perpeny dicular 00; the index will then cut latitude 73 on the quadrant.' i

Middle latitude sailing is performed by .bringing the distance. in casting or Westing,

. commonly called departure, to thel side A,

which is assumed as. the equator, thensetting vthe index to middle latitude;1.the perpen-- dicular from theend'of the distance or de. part-ure traced down willI then meet'the.dil-` ference of longitude on the index'. The in? dex should be set rather higher-than middle` latitude to give the-truc difference of longif tude.

y CASE 1.

Both latitudes and distance given, to dnd arture, and difference ot longitude` v y Example: AA ship in .latitude 55 sails between north. and easteiglity-six miles,l and has then come into latitud-'e' 556 l0 north. Required: her course, departure, and difference of longitude? i Rule Take the difference of latitude 70 on the side A; the distance 86 "on the index brought in contact with the perpendicular ot 70 will show the departure 50 on the side l, and the index without movingA will. show the course 35 .32 o'nth erquadrann '5T-helix' set the index-55 .35, talee fthedepartnr@ A150'4 onf-thc side A; assumed as the e'ql'1ator,'t1ace` its pe`rpeudicular down tothe index, upon which Two shipsjsai'ling v'bot-h, directly north or youwill find 88, the difference of longitude,-

required. y

CASE 2.

Both latitudes and difference of longitude given, to tind the course, distance, and departure.

Llrcmtplm Required: the course, distance, and departure between two places, one in latirude 5G l5 north and the other in latitude 58 35 north, and their difference of longitude being 20 30 Rule: Set the index to middle 57 25' and the difference of longitude being 150', (assuming 3 to L) take its' third 5() on the index, observe what perpendicular comes in contact with it; in this case it is 27 3 8l for the departure. Then Vthe differenceV of latitude being 14.0, take i ts"half 70 on the side A,-and the. depart-ure being 81, take its half 4:05` on the side B, set the index thereto, and the distance thereon doubled will be the required distance, 162. The index willalso show the course on the quadrant, nearly 30.

Mercator sailing is performed by taking the proper diierence of latitude, as also the meridian difference of latitude on the side-A.

From the end of the meridian 'difference of latitude take the di-fferenceof longitude on the side B,'to which set the index, and you willfhave the enlarged distance.A Then without moving the index you will iind the course by its edge on the quadrant. From theproper difference of latitude to the edge of the index will give the departure, and from .its

meeting with the index to the pivot willgivc' `the distance.

@Asn 1. Both latitudes and departure given, to find the course, distance, and difference of longisixtiei'h division'will quoteA 10G for the ine-- ridian part to that degree theleast moved up, and 10 Willgive 18 for meridian parts, added to 10G, will equatli; but as this runs beyond the end of the scale, takeits half, sixtytwo, and hall'7 seventy, equal thirty-live, and

h alf, fifty, yequal twentydive. rl`hen from 70 to the edge of the index will be S9, *rthe difference of longitude required.

The detached` index il in Fig. 2 and its use in connection with the scale and attached index in solving problems-in oblique-angled trigonometry will new be described. .The de tached indexcousists oa thin lath or strip having one edge graduated on"l both sides in the vsanne manner as the sides of the scale and theedge ofxthe attached index. At the end where'thegraduations commence is a circular piece'whose center is the zero-point or termination of the graduated edge. The circumference of the circular piece is graduated on v one side in points and quarter-points of the compass and on the other side in degrees. A2117 opening is made inthe circle, as shown at -P in Figs. 2 and 3 so as to leave the grad'uat'ed- I straight edge perfect andi clear and a point; to indicate the center of the circle. Its use is as follows, and as illustrated by the diagrams,

Figs. 4, 5, 6, 7, and 18 ofthe drawings: f

' Given threefsidesgs toind.the"'angle. Emple; Referring to Fjg. 4:.1 Given side' L M equals seventy,"side MAN equalsfft-y-f sevemandside' N 'L equals sixty-four.4 A Re-` quiredzthe angles? Rule:l Seti the center of the assisting -detached index to`70. on the side A,"(fo'"a base,) with its projecting leg over -the scale. Then take 57` the side M N on it,.and'take 6l', the. side N L on the indexof the scale, brin'gfthose numbers on each, respectively, in contact with each other. The angle L will be found indi-4 cated by the attached index on the quadrant as 4% points. 'lhenfor angle N hold the .xed index fast, set the center vof the de-.

tached index to 6i on the index of the scale, bring its Working-side to 70 on the side A, aud the circle will then show the angle N contained between the two indexes to be 6- points.

' CASE 2.

Two angles and a contained side given, to. iind the sides and angles. i

Example: Referring to Fig. 5..' Given thev side L M equals sixty; the angle L equals three i points; the angle M equals five points. Required: the side L N and 4side M N and the angle N? Rule: Set the attached index to three points from the side A on quadrant of scale and keep it there; also set the center of the assisting index toGO on the side A; bring it to make an angle of tive points with the side A. The intersection of the indexes will then indicate the side L if to b e 50 and the side M N to be To obtain the angle N, hold the index of scale fast on three points;` setthc center of the assisting index to 50 on it.;. b'ring its Working-side to (30 ou side A 0f scale. The angle N contained between both indexes will be eight points.

' CASE 3.

' Two sides and an angle opposite one of them given, to ind'the other angles and third side.'

Example: Referring to 'Fig'. 6. `Given the side M N equals seventy-seven, the side L M equals ninety, and the angle L equals five points. Required: the angles M and N and the side L N? Rule: Set the center of the assisting detached index on the attached index at 90;

make therewith the given angle five points.

Preserving'the angle, move the attached index to 0r from until its Working side comes in Contact with 77, the given side, on the side A on the scale. The assisting detached index will then show the quantity of the sidesought to be 65. On the quadrant of scale at the edgepof the attached index the angle M will be found to equal four points. angle N reverse thc assistingdetached index, place its center to 77 on the side A, bring (S5 (the side found on it) in contact with 90 on attached index, and the circle of the assisting index will then show on the side A that the angle N is equal to 7 points.

Example 2: Referring to Fig. 7. Given the side L M equals fifty, the side M N equals sixty, and the angle L equals four and onehalf points. Required: the angles M and N and the side L N2.

Rule.' Make the angle L four and one-half points with the attached index on the quadrant of scale and keep the index there. To 50 on the attached index set the center of the assisting detached index, bring 60 on the said .assisting index to the side A, and the assist-- ing index will then cut at 7871,-, which is the side sought. It will'also show the quantity of angle M eightpointson its circle. Reverse the CASE 4.

Two sides and the contained angle given, to find the other angles and third side.

To obtain the Example: Referring to Fig. S. Given side L M equals seVeniy-six,side LN equals forty, and the'contained angle equals three points. Required: the angles M and N andthe side M N2.

Rule: Set the attached 'index to 3 points from side A on quadrant of scale, the given angle, and hold it there; `Set the center ofassisting index to 40 (one of the given sides) on the attached index, bring the working edge of it to 76 on the side A, and the angle N included within both indexes will then be 10- points. show the third side to be 48. Reverse the assisting index, place its center to 76 on the side A, bring itsworking edge to 40 on index of scale held on the given angle, and the the squares, and also provided with an index F, marked as described and pivoted'to t'he square in the Vertex of the angle agreeing with the quadrant, and otherwise made'as herein shown and described.

ALEXANDER M. cHIsnoLM.

Witnesses:

HUGH M. DONALD, W. A. HENRY.

The assisting detached Vindex Will 

